If one is not interested in asymptotics of the variance, but in good enough bounds that yield almost-all results on \(\alpha _y\)  in short intervals, then very strong results were obtained by Friedlander and Iwaniec [1]}. By bounding the variance of sieve weights in short intervals [1]} they deduced that, if \(y \le X^{1/20}\)  and \(H/\log y \rightarrow \infty \)  then \(\sum _{n \in [x,x+H]}\alpha _y(n)\asymp HP_y\)  for almost all \(x \in [X,2X]\)  [1]}.
